If \(x\)  is an \(L_\infty (n_\theta )\) -affiliated operator on \({L}(L_2(n_\theta ))\) , its polar decomposition [1]} takes the form \(x=u|x|\) , where \(u\)  is a partial isometry in \(L_\infty (n_\theta )\) .
Moreover, the equality \(x^*=u|x|u^*\)  (see, e.g., [2]}) and Hölder's inequality show that
\(x\in L_{p}(n_\theta ) \Longrightarrow x^*\in L_p(n_\theta )\ \text{and}\ \Vert x^*\Vert _{L_p}=\Vert x\Vert _{L_p}.\) 
